Reasoning in the classroom
Jigsaw
5 jigsaw
Year 7
7 by 5 jigsaw
Support materials for teachers
Year 7 Reasoning in the classroom – Jigsaw
5 by 5 jigsaw
7 by 5 jigsaw
These Year 7 activities encourage learners to use their spatial reasoning
to solve problems.
Jigsaw
Learners explain and continue spatial patterns.
Includes:
■■ Jigsaw questions
■■ Markscheme
Corner, edge, inside
They use structure to solve a problem relating
to a jigsaw.
Includes:
■■ Explain and question – instructions for teachers
■■ Whiteboard – Corner, edge, inside
Fraction jigsaw
They solve a problem from the NRICH website, then create their own fraction,
decimal and percentage jigsaw.
Includes:
■■ Explain and question – instructions for teachers
Reasoning skills required
Identify
Communicate
Review
Learners select the appropriate
mathematics and determine
which techniques to use.
They present their work clearly,
using algebraic notation if
appropriate.
They review their work and
consider whether their findings
are accurate.
Procedural skills
Numerical language
■■ Multiplication
■■ Equivalent (Activity 3)
■■ Equivalent fractions, decimals and
percentages (Activity 3)
■■ Simple fraction addition and subtraction
(Activity 3)
Year 7 Reasoning in the classroom: Jigsaw
Introduction
Jigsaw
Activity 1 – Jigsaw
or
5 by 5 jigsaw
7 by 5 jigsaw
Outline
Learners use their spatial awareness to explain and extend numerical
patterns.
If needed, the jigsaw could be simplified for those with visual
impairment.
You will need
Q
Jigsaw questions
One page for each learner
M
Markscheme
Year 7 Reasoning in the classroom: Jigsaw
Activity 1 – Jigsaw – Outline
Q
5 by 5 jigsaw
7 by 5 jigsaw
Ali uses the same design to make a 9 by 5 jigsaw.
Explain how you know there are 45 pieces in his jigsaw.
1m
How many of each piece does Ali use in his 9 by 5 jigsaw?
red corner pieces =
brown side pieces =
blue side pieces =
yellow pieces =
purple pieces =
white pieces =
4m
Jigsaw
Activity 1 – Jigsaw – Questions
M
Activity 1 – Jigsaw – Markscheme
Q
Marks
i
1m
Answer
Justifies 45 by referring to 9 × 5, e.g.
●
5 lots of 9
●
Width × height
Or
Justifies 45 by referring to the pattern of
adding 10, e.g.
●
●
It’s 7 × 5 but with two more lots of 5 added
on
Each time you add 10
Or
If the six values in the next question part sum
to 45, justifies 45 by referring to the sum of
those values, e.g.
●
ii
4m
I worked them all out and counted them up
Gives all six correct values, i.e.
red = 4
brown = 10
blue = 10
yellow = 8
purple = 10
white = 3
Or 3m
Any five correct
Or 2m
Any four correct
Or 1m
Any two or three correct
Year 7 Reasoning in the classroom: Jigsaw
Activity 1 – Jigsaw – Markscheme
M
Activity 1 – Jigsaw – Exemplars
each colmn has 5 pieces and there are
severn colmns and 9 by 5 would have
nine colmns witch has ten more pieces
5 x 7 = 35
35 + 10 = 45
Part i: refers to adding 10; 1 mark
How many of each piece does Ali use in his 9 by 5 jigsaw?
10
red corner pieces =
4
blue side pieces =
10
yellow pieces =
8
purple pieces =
10
white pieces =
3
brown side pieces =
5 lots of 5 is 25 and 7 lots of 5 is 35
so I know 9 lots of 5 is 45
so it is right
How many of each piece does Ali use in his 9 by 5 jigsaw?
red corner pieces =
4
brown side pieces =
10
blue side pieces =
10
yellow pieces =
6
purple pieces =
10
white pieces =
3
I know because I worked out
4, 14, 24, 34, 42, 45 so it is 45
because that’s what I got
Part ii: all six correct values; 4 marks
Part i: refers to 9 × 5; 1 mark
Part ii: five correct values; 3 marks
●
As the sum of the values in part ii is not 45, this learner should
have realised that one or more of the values must be incorrect.
Part i: values sum to 45; 1 mark
How many of each piece does Ali use in his 9 by 5 jigsaw?
●
red corner pieces =
4
brown side pieces =
10
blue side pieces =
10
yellow pieces =
8
purple pieces =
10
white pieces =
3
This learner has understood the connection between the two
question parts so has added the values (cumulatively).
Part ii: all six correct values; 4 marks
I know because I added up the numbers below
Part i: values sum to 45; 1 mark
How many of each piece does Ali use in his 9 by 5 jigsaw?
●
red corner pieces =
4
brown side pieces =
11
blue side pieces =
9
yellow pieces =
8
purple pieces =
10
white pieces =
3
There are 45 pieces because
in a 9 by 5 jigsaw puzzle the will be
45 pieces
How many of each piece does Ali use in his 9 by 5 jigsaw?
red corner pieces =
10
brown side pieces =
1
blue side pieces =
10
yellow pieces =
1
purple pieces =
6
white pieces =
2
Year 7 Reasoning in the classroom: Jigsaw
Even though some of their values are incorrect, they do sum
to 45.
Part ii: four correct values; 2 marks
Part i: restatement of the question; 0 marks
Part ii: only one correct value; 0 marks
Activity 1 – Jigsaw – Exemplars
Corner, edge, inside
Activity 2 – Corner, edge, inside
or
Outline
This activity is designed to carry on from Activity 1 – Jigsaw.
Learners are given the number of pieces in a jigsaw and its dimensions.
They use this information to find the number of corner, edge and inside
pieces within the jigsaw.
The activity requires them to think algebraically, even if not using formal
algebraic notation.
You will need
WB
Year 7 Reasoning in the classroom: Jigsaw
Whiteboard – Corner, edge, inside
Activity 2 – Corner, edge, inside – Outline
Activity 2 – Corner, edge, inside
Show Corner, edge, inside on the whiteboard, and say that this picture is of a completed
jigsaw, but we can’t see the individual pieces.
Tell learners that the completed jigsaw has 35 pieces across and 30 pieces up.
Explain
Ask them to work in pairs or small groups to find how many of each type of piece (corner,
edge and inside) there are.
(Solution: 4 corner pieces; 33 + 33 + 28 + 28 = 122
edge pieces; 35 × 30 − 4 − 122 = 924 inside pieces.)
Or
Simplify by saying that the jigsaw
has 10 pieces across and 8 pieces
up. This allows learners to use a
drawing to support their reasoning.
■■ How are you finding the number of edge pieces? What relationship does this number
have to the perimeter of the jigsaw? Can you explain why?
(It is 8 less than the perimeter, because each corner piece has 2 edges which are not
counted.)
Question
■■ How are you finding the number of inside pieces? Can you think of more than one way
to work it out? How can you use area to help you work out the number of pieces? (The
inside pieces form a rectangle that is 33 by 28; 33 × 28 = 924.)
■■ Have you checked your answers? How?
■■ Can you tell me how you could work out the number of edge and inside pieces for any
size jigsaw? How could you use algebra to help you explain your rules?
Extension
■■ Tell learners that on a box it says that there are 192 jigsaw pieces altogether. The
dimensions of the completed jigsaw are 48cm by 36cm. (You may wish to tell learners
that jigsaw pieces are ‘square’.)
How many of each type of piece (corner, edge and inside) are there?
(Hint, but provide only if needed: Think about the factor pairs of 192.)
(Solution: 48cm by 36cm is in the ratio 4 : 3, so the number of pieces across : the number of
pieces up must also be in the ratio 4 : 3.
The only integers that multiply to 192 in this ratio are 16 and 12, so there must be 16 pieces
across and 12 pieces up. That gives 4 corner, 48 edge and 140 inside pieces.)
Year 7 Reasoning in the classroom: Jigsaw
Activity 2 – Corner, edge, inside – Explain and question
WB
Jigsaw
Activity 2 – Corner, edge, inside – Whiteboard
Fraction jigsaw
Activity 3 – Fraction jigsaw
or
Outline
This activity is designed to carry on from Activity 1 – Jigsaw and
Activity 2 – Corner, edge, inside but can be used independently. It focuses
on an NRICH activity which learners can access directly or the teacher can
print for them.
Learners use their knowledge of equivalent fractions to solve a jigsaw, then
they create their own fraction, decimal and percentage jigsaw for others to
solve.
You will need
The worksheet and instructions
at www.nrich.maths.org/5467
Scissors
Year 7 Reasoning in the classroom: Jigsaw
Activity 3 – Fraction jigsaw – Outline
Activity 3 – Fraction jigsaw
Give each group a copy of the worksheet found at
www.nrich.maths.org/5467
Explain
Explain that pieces fit together only if the edges that
touch contain fractions that are equivalent. Ask learners to
cut out the pieces and create the finished fraction jigsaw.
When completed, learners create their own jigsaw, using
their knowledge of equivalent fractions, decimals and
percentages, which they then give to other groups to
solve.
■■ How do you know when fractions are equivalent?
■■ If I asked you to write down every fraction that is equivalent to ½, how long would it
Question
take you? Why?
(Forever: there is an infinite number!)
■■ Some people think that when you add fractions you add the numerators and add the
denominators, e.g. they think that ⅔ + ⅕ = ⅜.
How would you convince them that this method is incorrect?
■■ How can you change fractions to decimals? Or decimals to percentages? Or . . . ?
Which is easiest, and why?
Year 7 Reasoning in the classroom: Jigsaw
Activity 3 – Fraction jigsaw – Explain and question